This is a collection of papers read at an international logic colloquium held at Oxford in 1963. The first half contains articles on intuitionistic and modal logics, the propositional calculus, and languages with infinitely long expressions by such logicians as Kripke, Bull, Harrop, and Tait. The second part is primarily concerned with recursive functions and features a monograph by Crossley on constructive order types, as well as contributions by Goodstein, Schütte, and Wang, among others. Especially noteworthy is Kripke's paper which (...) applies Cohen's methods to the semantics of intuitionistic logic.—P. J. M. (shrink)

Any logical analysis – and therefore philosophy itself – begins with the question: is there an indubitable knowledge? To answer this question Bertrand Russellappeals to tradition – both mathematical and philosophical – in which he recognizes himself. The process used by Russell, in order to build mathematics on a new foundation, was to build concepts logically, starting from atomic elements and known relationships. For example, a logical construction as that of the “class” is that a sentence, which includes a fixed (...) concept, is turned into a logically equivalent statement (example analyzed by Russell is “Scott” and “the author of Waverley”) which unlike the first, entails relations that include concepts. Thus, the primitive concept can be replaced and removed. The new method was named the “theory of types” and became the intellectual program of Russell and his successors. Among the method’s performances is solving logical and mathematical paradoxes. The importance of Russell’s logical investigations exceeded the reductionist program, stimulating the formalist program of philosophy andmethodology and the logical analysis of the language of science with a logical-mathematical apparatus. This program was definitively concluded, it seems, by Kurt Gödel’s results on the decidability problem in the formal axiomatic systems of P type, a problem closely linked to the question of “truth” and the two other meta-theoretical properties of P formal systems, namely “consistency” and “completeness”. Gödel’s Theorems do not affect the now established legitimacy of a meta-theoretical system concerning the formalizing processes, but prove that the hilbertian original program to reduce mathematics to logic is old and the very meaning of the whole process of realizing it must be analyzed again. Considering that in a formal P system, the decision process is demonstrability of true statements, Gödel’s research findings prove the consubstantial relationship of the four meta-theoretical properties Truth-Consistency-Completeness-Demonstrability. (shrink)

Resumen: Este artículo busca observar las diferencias entre el modelo sistémico funcional y el modelo pragmático-formal en su comprensión de la experticia científica y su rol en las sociedades modernas. Se elaborará un breve diagnóstico acerca de la importancia de la confianza en experticia científica en la sociedad contemporánea y cómo este proceso ha sido analizado. Luego, se analizará la descripción del conocimiento científico en la sociedad contemporánea desde el modelo sistémico funcional. Utilizando los conceptos de diferenciación funcional y (...) acoplamiento estructural, se sugerirá que, desde este modelo, el conocimiento científico es una forma de comunicación especializada propia de un sistema funcional de la sociedad. A continuación, se describirá la aproximación del modelo pragmático formal. Especial importancia tienen aquí las ideas de mundo de la vida y esfera pública. Sobre la base de estas caracterizaciones, se comparan las visiones de estos autores y se analizan las diferencias en su conceptualización de la experticia. Se argumenta que las diferencias de estos dos modelos en relación con el estatus ontológico del mundo de la vida marcan sus visiones de la experticia y los debates actuales acerca de sus posibilidades de convertibilidad en la literatura especializada. El artículo finaliza con un resumen y posibles líneas futuras de investigación.: This article seeks to observe the differences between the functional-systemic model and the pragmatic-formal model in their understanding of scientific expertise and its role in modern societies. First, a brief diagnosis about the importance of trust in scientific expertise in contemporary society and how this process has been analyzed in the specialized literature will be made. Then, the description of scientific knowledge will be analyzed from the point of view of the functional systemic model. Using the concepts of functional differentiation and structural coupling, it will be argued that scientific knowledge is here considered a specialized form of communication characteristic of a functional system. Next, the approximation of the formal-pragmatic model is described. Of special importance here are the ideas of lifeworld and public sphere. On the basis of these characterizations, the visions of these authors are compared and the differences are analyzed in relation to their conceptualization of expertise. It is then suggested that the differences between these two models in relation to the ontological status of the lifeworld shape their views of scientific expertise as well as current debates on this topic. The article ends with a summary and possible future line of research. (shrink)

This is a very fine elementary-to-intermediate level text of mathematical logic. The initial chapter of the book consists of a good discussion of standard topics in modern formal logic including arguments and argument forms, logical functors, validity, proof, the axiomatic method, interpretations, and logical systems. The book then proceeds in subsequent chapters to a development of increasingly rich systems of sentential logic, systems of natural deduction, and a chapter on consistency and completeness of formal (...) class='Hi'>systems. This takes the student through approximately one half of the book. The second half is devoted to more advanced topics such as non-standard systems of propositional logic, and predicate logic of the first order. Finally, there are chapters on higher functional calculi, the logical paradoxes, and non-standard functional systems. A good deal of attention is paid throughout to completeness and consistency properties, the decision problem for various systems, and algebraic structure. There are no frills or "gimmicks" in this book, but good intelligent discussions of important mathematical issues, and many theorems which are left as exercises to the student. Some items which one wishes had been included are normal forms, and elementary discussions of the theories of recursion and logical models.—H. P. K. (shrink)

This is a first attempt to formalize the language required for analysis of purposive organizations or systems into the subordinate systems of which they organically consist. The authors take a philosophic position midway between Atomism and the Absolute; like Aristotle, they take a finite, complex individual as the ultimate referent of explanation. The sole primitive is "s///tOx," interpreted as "upon analytic dissection, the system t organized by [the property] x." It is claimed without argument that the relationship is (...) independent of who, so to speak, is doing the dissecting. This paper is welcome as a first attack on a difficult problem, although it does not pay off in the production of non-trivial theorems or in clarification much beyond that afforded by technical English.--N. D. B. Jr. (shrink)

Pursuing Justice, Second Edition, examines the issue of justice by considering the origins of the idea, formal systems of justice, current global issues of justice, and ways in which justice might be achieved by individuals, organizations, and the global community. Part 1 demonstrates how the idea of justice has emerged over time, starting with religion and philosophy, then moving to the justice as a concern of the state, and finally to the concept of social justice. Part 2 outlines (...) the very different mechanisms used by various nations for achieving state justice, including systems based on common law, civil law, and Islamic law, with a separate discussion of the US justice system. Part 3 focuses on four contemporary issues of justice: war, genocide, slavery, and the environment. Finally, Part 4 shows how individuals and organizations can go about pursuing justice, and describes the rise of global justice. This updated timely book helps students understand the complexities and nuances of a society's pursuit of justice. It provides students with the foundations of global justice systems, integrating Greek philosophies and major religious perspectives into a justice perspective, and contributes to undergraduate understanding of international justice bodies, NGOs, and institutions. New edition is completely updated and revised to achieve relevance for today's students Covers concepts of justice as well as ideas for pursuing and achieving justice Examines how our modern laws began, and traces their evolution to today's laws Presents concepts and issues in justice studies as well as a comparison of several systems of law Teaching resources include discussion questions and real-world examples. (shrink)

A number of systems of logic, including formulations of the two-valued propositional logic, alethic modal propositional logics and deontic modal propositional logics, are described and discussed with the aim of utilizing some of these systems for the sociological analysis of normative structures. Evidence is adduced that certain normative concepts are explicated or partially explicated in these systems.--W.T.

The paper presents, in the synthetic way, the issue of the rationality of the tax system and taxation principles in relation to the clearly visible fiscal crisis in the 21st century caused by, among others, COVID-19 pandemic. The analysis which is theoretical (descriptive, comparative) was carried out by using tools indicated in the area of New Institutional Economics (NIE). Attention was devoted primarily to the importance of specific institutional arrangements, broadly understood as the “rules of the game” applicable in the (...) political and economic spheres, and related to the problems of public finances. Research has shown that the appropriate design of tax rules may contribute, to some extent, to limiting the fiscal crisis. Tax policy, and even more broadly, fiscal policy, may determine the formula in which the economic achievements regarding compliance with tax principles and fiscal rules can be implemented. What is primarily important is the legal framework (one of the most important formal institutions) within which state policy in the field of public finances should be implemented. (shrink)

Formal dialectic has its roots in ancient dialectic. We can trace this influence in Charles Hamblin’s book on fallacies, in which he introduced his first formal dialectical systems. Earlier, Paul Lorenzen proposed systems of dialogical logic, which were in fact formal dialectical systems avant la lettre, with roles similar to those of the Greek Questioner and Answerer. In order to make a comparison between ancient dialectic and contemporary formal dialectic, I shall formalize (...) part of the Aristotelian procedure for Academic debates. The resulting system will be compared (1) with Van Eemeren and Grootendorst’s system of rules of Critical Discussion (the pragma-dialectical discussion procedure), which must, however, first itself be reconstructed as a formal dialectical system, and (2) with a Hamblin-type system, and (3) a Lorenzen-type system. When drawing comparisons, it will become clear that there is a line to be drawn from Aristotle to formal dialectic and pragma-dialectics, extending to contemporary computational models of argument. (shrink)

This paper examines whether classical extensional mereology is adequate for formalizing the whole–parts relation in quantum chemical systems. Although other philosophers have argued that classical extensional and summative mereology does not adequately formalize whole–parts relation within organic wholes and social wholes, such critiques often assume that summative mereology is appropriate for formalizing the whole–parts relation in inorganic wholes such as atoms and molecules. However, my discussion of atoms and molecules as they are conceptualized in quantum chemistry will establish that (...) standard mereology cannot adequately fulfill this task, since the properties and behavior of such wholes are context-dependent and cannot simply be reduced to the summative properties of their parts. To the extent that philosophers of chemistry have called for the development of an alternative mereology for quantum chemical systems, this paper ends by proposing behavioral mereology as a promising step in that direction. According to behavioral mereology, considerations of what constitutes a part of a whole is dependent upon the observable behavior displayed by these entities. Thus, relationality and context-dependence are stipulated from the outset and this makes behavioral mereology particularly well-suited as a mereology of quantum chemical wholes. The question of which mereology is appropriate for formalizing the whole–parts relation in quantum chemical systems is relevant to contemporary philosophy of chemistry, since this issue is related to the more general questions of the reducibility of chemical wholes to their parts and of the reducibility of chemistry to physics, which have been of central importance within the philosophy of chemistry for several decades. More generally, this paper puts contemporary discussions of mereology within the philosophy of chemistry into a broader historical and philosophical context. In doing so, this paper also bridges the gap between formal mereology, conceived as a branch of formal ontology, and “applied” mereology, conceived as a branch of philosophy of science. (shrink)

Common sense is on the one hand a certain set of processes of natural cognition - of speaking, reasoning, seeing, and so on. On the other hand common sense is a system of beliefs (of folk physics, folk psychology and so on). Over against both of these is the world of common sense, the world of objects to which the processes of natural cognition and the corresponding belief-contents standardly relate. What are the structures of this world? How does the scientific (...) treatment of this world relate to traditional and contemporary metaphysics and formal ontology? Can we embrace a thesis of common-sense realism to the effect that the world of common sense exists uniquely? Or must we adopt instead a position of cultural relativism which would assign distinct worlds of common sense to each group and epoch? The present paper draws on recent work in computer science (especially in the fields of naive and qualitative physics), in perceptual and developmental psychology, and in cognitive anthropology, in order to consider in a new light these and related questions and to draw conclusions for the methodology and philosophical foundations of the cognitive sciences. (shrink)

The grammatical paradigm used to be a model for entire areas of cognitive science. Its primary tenet was that theories are axiomatic-like systems. A secondary tenet was that their predictions should be tested quickly and in great detail with introspective judgments. While the grammatical paradigm now often seems passé, we argue that in fact it continues to be as efficient as ever. Formal models are essential because they are explicit, highly predictive, and typically modular. They make numerous critical (...) predictions, which must be tested efficiently; introspective judgments do just this. We further argue that the grammatical paradigm continues to be fruitful. Within linguistics, implicature theory is a recent example, with a combination of formal explicitness, modularity, and interaction with experimental work. Beyond traditional linguistics, the grammatical paradigm has proven fruitful in the study of gestures and emojis; literature (“Free Indirect Discourse”); picture semantics and comics; music and dance cognition; and even reasoning and concepts. We argue, however, that the grammatical paradigm must be adapted to contemporary cognitive science. Computational methods are essential to derive quantitative predictions from formal models (Bayesian pragmatics is an example). And data collection techniques offer an ever richer continuum of options, from introspective judgments to large-scale experiments, which makes it possible to optimize the cost/benefit ratio of the empirical methods that are chosen to test theories. (shrink)

The tools of logic are used properly or improperly relative to two interrelated purposes. Logic is both a symbolism for the expression of the formal structures of thought and an inference mechanism. Formalization in philosophical logic is justified to the extent that it contributes to our understanding of logical properties and the conceptual problems they may help to state, clarify, or resolve. This view of the value and limits of formalization in logic affords a pragmatic perspective that in principle (...) should selectively support the development of particular formal systems, while excluding others as unjustified. Yet the pragmatic grounds are so liberal as to disallow virtually no exercises in formalization as entirely useless. Logic is abused in another sense, when it is wrongly used, not by violating logical canons generally, but by presupposing that substantive metaphysical, normative, or scientific content can be derived from purely formal logical relations. (shrink)

The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible with (...) successful representation, scientists often rely on the existence of a ‘mature mathematical formalism’, where the latter refers to a—mathematically formulated and physically interpreted—notational system of locally applicable rules that derive from (but need not be reducible to) fundamental theory. As mathematical formalisms undergo a process of elaboration, enrichment, and entrenchment, they come to embody theoretical, ontological, and methodological commitments and assumptions. Since these are enshrined in the formalism itself, they are no longer readily obvious to either the novice or the proficient user. At the same time as formalisms constrain what may be represented, they also function as inferential and interpretative resources. (shrink)

Fichte’s Foundations of the Entire Wissenschaftslehre 1794 is one of the most fundamental books in classical German philosophy. The use of laws of thought to establish foundational principles of transcendental philosophy was groundbreaking in the late eighteenth and early nineteenth century and is still crucial for many areas of theoretical philosophy and logic in general today. Nevertheless, contemporaries have already noted that Fichte’s derivation of foundational principles from the law of identity is problematic, since Fichte lacked the tools to correctly (...) present the formal parts of Foundations. In this paper, however, we argue that Fichte’s approach intuitively offers an important contribution to transcendental philosophy, and especially to philosophy of logic. We first point out the difficulties of Fichte’s logic in the Foundations and improve it in a second part on the basis of a formal system in which both propositional logic and syllogistic are combined. (shrink)

“Formal logic”, an expression created by Kant to characterize Aristotelian logic, has also been used as a name for modern logic, originated by Boole and Frege, which in many aspects differs radically from traditional logic. We shed light on this paradox by distinguishing in this paper five different meanings of the expression “formal logic”: (1) Formal reasoning according to the Aristotelian dichotomy of form and content, (2) Formal logic as a formal science by opposition to (...) an empirical science, (3) Formal systems in the sense of Hilbert, Curry and the formalist school, (4) Symbolic logic, a science using symbols, such as Venn diagrams, (5) Mathematical logic, a mathematical approach to reasoning. We argue that these five meanings are independent and that the meaning (5) is the one which better characterized modern logic, which should therefore not be called “formal logic”. (shrink)

Ontologies are some of the most central constructs in today's large plethora of knowledge technologies, namely in the context of the semantic web. As their coinage indicates, they are direct heirs to the ontological investigations in the long Western philosophical tradition, but it is not easy to make bridges between them. Contemporary ontological commitments often take causality as a central aspect for the ur-segregation of entities, especially in scientific upper ontologies; theories of causality and philosophical ontological investigations often go (...) hand-in-hand, and were essentially inseparable in medieval thought. This constitutes the foundation for a bridge, and this article analyzes the causality-based ontology of the late medieval philosopher Dietrich of Freiberg from the viewpoint of today's upper-ontology engineering. In this bridging attempt, it offers a translation into English of the first part of Dietrich's De origine (abbreviated title) that is a compromise between traditional scholarly translations of medieval Latin philosophical texts and contemporary ontology. (shrink)

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to (...) that formalization purpose, of some modal schemes, such as S4 and S5. It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon. (shrink)

The author has constructed a concept of conditionals by synthetizing and developing unconnected insights scattered through the literature. The result is incorporated in a formal deductive system, based on a series of "paradox-free" systems initiated by Alonzo Church and interpreted according to principles suggested chiefly by Everett Nelson and by Anderson and Belnap. The basic concept is the sufficiency relation holding between clauses of a conditional, or rather between the relevant states of affairs asserted by the clauses. The (...) logic of sufficiency is developed by using a phenomenological method, much like that of the ordinary-language linguists, to place restrictions on truth-functional logic. For example, conjunction is replaced by adjunction [ = df. ~ ] and this concept is used to modify modus ponens and simplification. Relevance requirements avoid the paradoxes of material implication. These principles, together with a number of physical modalities and some modifications of the concept of induction, are used to attack Goodman's paradox and the paradoxes of confirmation, and to form a concept of cause. The formal system incorporates the principles judged desirable by the analysis of how conditionals are used in discourse. Mr. Barker is modest in his claims, emphasizing that his material is not original and saying only that some of the famous paradoxes "show signs of yielding." Interested students will be grateful for this monograph. It is a valuable compendium of widely scattered work of a difficult and complicated subject, and should be both helpful and stimulating. Moreover, it is particularly clear and readable.--L. G. (shrink)

The perception of separation between school and home/community is related to diminished achievement in school and lack of motivation to learn STEM subjects. The National Council of Educational Research and Training is among many research organisations that have strongly recommended that schools bridge the disconnect between school-based knowledge and learners’ everyday knowledge. We designed the SPIRALS curriculum to bridge this gap between formal science and students’ everyday lives. SPIRALS helps students explore community-based practices to learn about science, environmental sustainability (...) and systems thinking. We implemented the SPIRALS curriculum in a private, urban, English medium school in Western India with approximately 315 students and their four teachers, 214 of whom also participated in the research from which our conclusions are drawn. Our findings about program impacts rely upon analysis of interviews with teachers and students, as well as student work, and conference participation assessment surveys distributed after a capstone experience at which students present their work. This chapter describes our findings about how students learned science, environmental sustainability, and systems thinking through engagement with community-based practices. We also discuss the process of how the SPIRALS approach worked in India and how it could be expanded into a broader learning model across different socio-cultural contexts within India. (shrink)

Contemporary logicians have expanded upon the old notions of psychologism in logic and proposed new, weakened versions of it. Those weakened versions postulate that psychologistic logic does not have to inform about the ontology or metaphysics of reasoning. Instead, logic applied in cognitive science could serve as one of many paradigms for making empirical predictions about the observable process of human reasoning. The purpose of this article is to entertain this notion and answer the question: what properties should a (...) logical system formally representing actual human reasoning have? Based on the existing evidence from cognitive science and neuroscience we identified three potential candidates: context-sensitivity (satisfied for example by adaptive logics), content-sensitivity (satisfied by non-Fregean logics) and probabilism (satisfied for example by fuzzy logics). (shrink)

This paper disucsses the Boolean algebraic axiomatic system of situations suggested by the Polish logician Roman Suszko (1919-1979). The paper will specifically examine the adequacy of the axioms, definitions and theorems of Suszko’s system as a model for Ludwig Wittgenstein’s Tracatus Logico-Philosophicus. It will be shown how the formal properties of Suszko’s system - the atomicity and completeness of the Boolean algebraic system - can be employed in order to clarify key concepts of the situational part of the Tractarian (...) ontology. After considering the formal reconstruction of the Tractarian concepts of teh world and logical space, a controversial issue pertaining to necessary facts in the Tracatus will be addressed. This will be followed by a formal clarification of the Tractarian concepts of logical place and possible worlds, the latter being identified as combinations of states and affairs, which are, according to the Tractarian ontology, the simplest kinds of situations. (shrink)

Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present (...) in these formal constructions and discusses them as an argument for the alternative semantic and propositional-structure accounts of the applicability of mathematics. (shrink)

Gibson developed the affordance concept to complement his theory of direct perception that stands in sharp contrast with the prevalent inferential theories of perception. A comparison of the two approaches shows that the distinction between them also has an ontological aspect. We trace the history and newer formalizations of the notion of affordance and discuss some competing opinions on its scope. Next, empirical work on the affordance concept is reviewed in brief and the relevance of dynamical systems theory to (...) affordance research is demonstrated. Finally, the striking but often neglected convergence of the ideas of Gibson and those of certain Continental philosophers is discussed. (shrink)

The logical theories of Stanislaw Leśniewski differ profoundly form classical formal systems. Unlike the latter, they do not have an entirely predetermined vocabulary. Nor do they have a determined list of functors of syntactical-semantical categories. Due to formalized directives for definitions, the logics of Leśniewski are constructed progressively, making new theses and consequently functors of new syntactical-semantical categories accesible. In this article we present the genetic aspect associated with these theses-definitions. We also show that the property of creativity (...) makes it possible to bridge some of the fundamental gaps in contemporary classical logics. (shrink)

The logical theories of Stanislaw Leśniewski differ profoundly form classical formal systems. Unlike the latter, they do not have an entirely predetermined vocabulary. Nor do they have a determined list of functors of syntactical-semantical categories. Due to formalized directives for definitions, the logics of Leśniewski are constructed progressively, making new theses and consequently functors of new syntactical-semantical categories accesible. In this article we present the genetic aspect associated with these theses-definitions. We also show that the property of creativity (...) makes it possible to bridge some of the fundamental gaps in contemporary classical logics. (shrink)

This thesis is on information dynamics modeled using *dynamic epistemic logic*. It takes the simple perspective of identifying models with maps, which under a suitable topology may be analyzed as *topological dynamical systems*. It is composed of an introduction and six papers. The introduction situates DEL in the field of formal epistemology, exemplifies its use and summarizes the main contributions of the papers.Paper I models the information dynamics of the *bystander effect* from social psychology. It shows how augmenting (...) the standard machinery of DEL with a decision making framework yields mathematically self-contained models of dynamic processes, a prerequisite for rigid model comparison.Paper II extrapolates from Paper I's construction, showing how the augmentation and its natural peers may be construed as maps. It argues that under the restriction of dynamics produced by DEL dynamical systems still falls a collection rich enough to be of interest. Paper III compares the approach of Paper II with *extensional protocols*, the main alternative augmentation to DEL. It concludes that both have benefits, depending on application. In favor of the DEL dynamical systems, it shows that extensional protocols designed to mimic simple, DEL dynamical systems require infinite representations. Paper IV focuses on *topological dynamical systems*. It argues that the *Stone topology* is a natural topology for investigating logical dynamics as, in it, *logical convergence* coinsides with topological convergence. It investigates the recurrent behavior of the maps of Papers II and III, providing novel insigths on their long-term behavior, thus providing a proof of concept for the approach.Paper V lays the background for Paper IV, starting from the construction of metrics generalizing the Hamming distance to infinite strings, inducing the Stone topology. It shows that the Stone topology is unique in making logical and topological convergens coinside, making it the natural topology for logical dynamics. It further includes a metric-based proof that the hitherto analyzed maps are continuous with respect to the Stone topology. Paper VI presents two characterization theorems for the existence of *reduction laws*, a common tool in obtaining complete dynamic logics. In the compact case, continuity in the Stone topology characterizes existence, while a strengthening is required in the non-compact case. The results allow the recasting of many logical dynamics of contemporary interest as topological dynamical systems. (shrink)

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements. It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.After the (...) discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions. Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all. (shrink)

It is the aim of the present study to introduce the reader to the ways of thinking of those contemporary philosophers who apply the tools of symbolic logic to classical philosophical problems. Unlike the "conti nental" reader for whom this work was originally written, the English speaking reader will be more familiar with most of the philosophers dis cussed in this book, and he will in general not be tempted to dismiss them indiscriminately as "positivists" and "nominalists". But the (...) English version of this study may help to redress the balance in another respect. In view of the present emphasis on ordinary language and the wide spread tendency to leave the mathematical logicians alone with their technicalities, it seems not without merit to revive the interest in formal ontology and the construction of formal systems. A closer look at the historical account which will be given here, may convince the reader that there are several points in the historical develop ment whose consequences have not yet been fully assessed: I mention, e. g., the shift from the traditional three-level semantics of sense and deno tation to the contemporary two-level semantics of representation; the relation of extensional structure and intensional content in the extensional systems of Wittgenstein and Carnap; the confusing changes in labelling the different kinds of analytic and apriori true sentences; etc. Among the philosophically interesting tools of symbolic logic Lesniewski's calculus of names deserves special attention. (shrink)

Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of (...) theorems with much wider scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction. (shrink)

Fred Sommers passed away in October of 2014 in his 92nd year. Having begun his teaching at Columbia University, he eventually became the Harry A. Wolfson Chair in Philosophy at Brandeis University, where he taught from 1963 to 1993. During his long and productive career, Sommers authored or co-authored over 50 books, articles, reviews, etc., presenting his ideas on numerous occasions throughout North America and Europe. His work was characterized by a commitment to the preservation and application of historical insights (...) and to the value of a well-articulated, coherent logical system. He was recognized for his independence and refusal to accept any view on the basis of authority alone. This made him a formidable critic but accounted in part for his many innovative and original ideas. In spite of his general contrariness in logic, Sommers earned the respect of the majority of his contemporaries, including Russell, Quine, van Benthem, Hacking, Suppes, and Strawson. In 2005, he was the subjec... (shrink)

What exactly is so appealing in formal science, such that its influence can be seen in numerous disciplines? This contemporary addiction to practical convenience in formal science has turned a blind eye to its other side. This book provides a way to understand the nature of formal science, in relation to systems theory for practical convenience.

Cognitive science is, more than anything else, a pursuit of cognitive mechanisms. To make headway towards a mechanistic account of any particular cognitive phenomenon, a researcher must choose among the many architectures available to guide and constrain the account. It is thus fitting that this volume on contemporary debates in cognitive science includes two issues of architecture, each articulated in the 1980s but still unresolved: " • Just how modular is the mind? – a debate initially pitting encapsulated mechanisms (...) against highly interactive ones. • Does the mind process language-like representations according to formal rules? – a debate initially pitting symbolic architectures against less language-like architectures. " Our project here is to consider the second issue within the broader context of where cognitive science has been and where it is headed. The notion that cognition in general—not just language processing—involves rules operating on language-like representations actually predates cognitive science. In traditional philosophy of mind, mental life is construed as involving propositional attitudes—that is, such attitudes towards propositions as believing, fearing, and desiring that they be true—and logical inferences from them. On this view, if a person desires that a proposition be true and believes that if she performs a certain action it will become true, she will make the inference and perform the action. (shrink)

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

This book is a translation of some of the more important parts of the Grundgesetze of Frege: the introduction, the first part of the first volume which gives an exposition of the construction, rules, axioms of Frege's formal system, and two appendices, one of which is from the second volume and gives Frege's analysis of the paradox found by Russell in his system. The editor has provided a long introduction "for those not familiar with Frege," although it will benefit (...) those who have something more than acquaintance with his name as well. Frege's original two-dimensional symbolism has been preserved, but the editor has provided enough discussion in his introduction so as to make the going easier. Next to a complete translation of the Grundgesetze, this work is the most useful introduction to the magnum opus of the nineteenth century's greatest logician.—P. J. M. (shrink)

This introductory logic text for philosophers stresses decision procedures for the propositional and predicate calculi. In the treatment of the former the authors first present the apparatus for constructing logical inferences and establishing their validity; they then reformulate the system in an axiomatic way and prove its consistency and completeness; they also discuss, for example, techniques for proving independence of axioms. The lower predicate calculus is then developed, first in a monadic form, then in an extended one for which decision (...) procedures are available. Although the authors introduce predicates of degree two or more along with multiple quantification, they do not prove this system of the full predicate calculus complete. The last chapter provides a well-articulated treatment of syllogistic in connexion with quantification theory. There are appendices on conjunctive normal forms and notation without parentheses. The answers to the exercises are at the end of the book.—P. J. M. (shrink)

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. (...) The problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a chain of arguments.From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs. (shrink)

Wilfried Sieg. Formal Systems, Church Turing Thesis, and Gödel's Theorems: Three Contributions to The MIT Encyclopedias of Cognitive Science.

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking just one (...) syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (shrink)

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. (...) The problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a chain of arguments.From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs. (shrink)

One of five short texts in the publisher's "Foundations of Logic Series." Fisk presents a sentential calculus and extensions to uniform and full first-order quantification in terms of natural-deduction principles. The principles laid down are continually justified by reference to our instinctive use of language. In keeping with this approach, Fisk is concerned to base the system on an intensional implication relation which will avoid the familiar paradoxes. Unfortunately, his system S can be proved equivalent to the classical two-valued calculus. (...) This unanticipated result proceeds from principle S7, " p → q ∴ →." However, if S7 is dropped, the philosophical claims made for S hold good. By virtue of short but incisive discussions of controversial philosophical issues, e.g., existence and subjunctive conditionals, this book is a contribution as well as an introduction to philosophical thinking.—J. B. B. (shrink)

The thesis presents and defends an essentialistic semantic theory. The theory has a formal and an informal component. The formal component is the intensional modal logic of Aldo Bressan, a possible worlds system that has certain advantages over better known systems like those of Carnap or Krip.

"Philosophical logic" describes two distinct areas: the investigation of the fundamental concepts of logic, the formal investigation of alternatives and extensions to classical logic. The first is a philosophical discipline, concerned with notions like truth, propositions, necessity, logical consequence, vagueness, and reasoning. The second is a technical discipline, devoted to developing formal logical systems-modal logics, second-order logics, intuitionistic logics, relevance logics, logics of vagueness and conditionals-and proving things about them. Most texts in philosophical logic focus on one (...) of these areas, but in this book John MacFarlane treats them together in an integrated way, showing how philosophical considerations motivate the technical projects, and how the constraints revealed by the technical projects illuminate the philosophical issues. Topics covered include quantifiers, modal logic, indicative conditionals, model-theoretic and proof-theoretic characterizations of logical consequence, intuitionistic logic, fundamental logical disagreement, relevance logic, the relation of logic and reasoning, and vagueness. Each chapter is organized around suggested readings and includes exercises. Key Features: An integrated treatment of the technical and philosophical issues comprising philosophical logic Written by a leading authority on logicism and logical form and a successful expositor and teacher Designed to serve students taking only one course in logic beyond the introductory level Provides tools and concepts necessary to understand work in many areas of analytic philosophy Includes exercises, suggested readings, and suggestions for further exploration in each chapter. (shrink)